Two-variable first order logic with modular predicates over words

Authors Luc Dartois, Charles Paperman

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Luc Dartois
Charles Paperman

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Luc Dartois and Charles Paperman. Two-variable first order logic with modular predicates over words. In 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 20, pp. 329-340, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)


We consider first order formulae over the signature consisting of the symbols of the alphabet, the symbol < (interpreted as a linear order) and the set MOD of modular numerical predicates. We study the expressive power of FO^2[<,MOD], the two-variable first order logic over this signature, interpreted over finite words. We give an algebraic characterization of the corresponding regular languages in terms of their syntactic morphisms and we also give simple unambiguous regular expressions for them. It follows that one can decide whether a given regular language is captured by FO^2[<,MOD]. Our proofs rely on a combination of arguments from semigroup theory (stamps), model theory (Ehrenfeucht-Fraïssé games) and combinatorics.
  • First order logic
  • automata theory
  • semigroup
  • modular predicates


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